A practical approach to lake water density from electrical conductivity and temperature

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electrical conductivity and temperature

Santiago Moreira, Martin Schultze, Karsten Rahn, Bertram Boehrer

To cite this version:

Santiago Moreira, Martin Schultze, Karsten Rahn, Bertram Boehrer. A practical approach to lake

wa-ter density from electrical conductivity and temperature. Hydrology and Earth System Sciences,

Eu-ropean Geosciences Union, 2016, 20 (7), pp.2975 - 2986. �10.5194/hess-20-2975-2016�. �hal-01587454�

www.hydrol-earth-syst-sci.net/20/2975/2016/ doi:10.5194/hess-20-2975-2016

© Author(s) 2016. CC Attribution 3.0 License.

A practical approach to lake water density from electrical

conductivity and temperature

Santiago Moreira1,2,3, Martin Schultze1, Karsten Rahn1, and Bertram Boehrer1

1_{UFZ-Helmholtz-Centre for Environmental Research, Department Lake Research, Brueckstrasse 3a, 39114 Magdeburg,} Germany

2_{Laboratoire des Sciences du Climat et de l’Environnement, LSCE/IPSL, CEA-CNRS-UVSQ, Université Paris-Saclay,} 91191 Gif-sur-Yvette, France

3_{Institute of Hydrobiology, TU-Dresden, Zellescher Weg 40, 01217 Dresden, Germany}

Correspondence to:Santiago Moreira (santiago.moreira-martinez@lsce.ipsl.fr)

Received: 22 January 2016 – Published in Hydrol. Earth Syst. Sci. Discuss.: 17 February 2016 Revised: 3 June 2016 – Accepted: 20 June 2016 – Published: 22 July 2016

Abstract. Density calculations are essential to study strat-ification, circulation patterns, internal wave formation and other aspects of hydrodynamics in lakes and reservoirs. Cur-rently, the most common procedure is the use of CTD (con-ductivity, temperature and depth) profilers and the conversion of measurements of temperature and electrical conductivity into density. In limnic waters, such approaches are of limited accuracy if they do not consider lake-specific composition of solutes, as we show. A new approach is presented to corre-late density and electrical conductivity, using only two spe-cific coefficients based on the composition of solutes. First, it is necessary to evaluate the lake-specific coefficients con-necting electrical conductivity with density. Once these coef-ficients have been obtained, density can easily be calculated based on CTD data. The new method has been tested against measured values and the most common equations used in the calculation of density in limnic and ocean conditions. The re-sults show that our new approach can reproduce the density contribution of solutes with a relative error of less than 10 % in lake waters from very low to very high concentrations as well as in lakes of very particular water chemistry, which is better than all commonly implemented density calculations in lakes. Finally, a web link is provided for downloading the corresponding density calculator.

1 Introduction

Density is one of main physical quantities governing the hy-drodynamics, stratification, and mixing in lakes and reser-voirs. Water quality in lakes is controlled by biological and biogeochemical processes which depend on the availability of oxygen in deep waters and nutrients in surface waters. Both phenomena are controlled by the duration and exten-sion of the turnover period, which is dependent on density gradients. Therefore, density is a very important variable in numerical models for the simulation of the behaviour of lakes under changing conditions, e.g. due to management mea-sures or phenomena related to global climate change.

The density of lake water (at atmospheric pressure) de-pends on temperature and dissolved water constituents. Since temperature, chemical composition and concentrations may vary over time, from lake to lake or even within a particular lake due to seasonal stratification or meromixis, numerical models of lakes calculate the density internally. There are several approaches to calculate water density in lakes. Most of them are general equations that do not always reflect spe-cific properties of lakes. If enough measurements of density for the relevant temperature range are available and compo-sition and concentrations of the main constituents are con-stant, regressions can be used to generate a mathematical for-mula for density in a specific lake (e.g. Jellison et al., 1999; Vollmer et al., 2002; Karakas et al., 2003). If the composi-tion is constant and the main constituents are ions, electrical conductivity or salinity may be used as an easy-to-measure

proxy for concentrations (Bührer and Ambühl, 1975; Chen and Millero, 1986; Pawlowicz and Feistel, 2012).

Imboden and Wüest (1995) discussed the influence of dis-solved substances on (potential) density because both the concentration and chemical composition of the total dis-solved solids change considerably from lake to lake (see e.g. Boehrer and Schultze, 2008). The effects of dissolved solids on density stratification have been studied in lake-specific in-vestigations in Lake Malawi (Wüest et al., 1996) and in Lake Matano (Katsev et al., 2010). In some cases, the specific con-tribution of ions such as calcium, carbonate or dissolved iron can control the permanent stratification in lakes such as in La Cruz (Spain) (Rodrigo et al., 2001), Cueva de la Mora (Spain) (Sanchez-España et al., 2009) or Waldsee (Germany) (Dietz et al., 2012).

Density of pure water can be calculated using mathemat-ical expressions such as those in Kell (1975) or Tanaka et al. (2001). Density calculations of natural waters require ad-ditional terms to include the contributions of dissolved sub-stances. Specific formulas have been developed for ocean conditions. The UNESCO equations developed by Fofonoff and Millard (1983) have been the standard for a long period. They used temperature and practical salinity based on electri-cal conductivity measurements. Because seawater conditions are a known reference and the approaches provide stable re-sults over a wide range of temperatures and electrical con-ductivity, these have been applied in limnic systems and im-plemented in numerical models such as DYRESM (Imberger and Patterson, 1981; Gal et al., 2009), ELCOM (Hodges and Dallimore, 2007), GOTM (Burchard et al., 1999; Umlauf et al., 2005) or CE-QUAL-W2 (Cole and Buchak, 1995). Re-cently the ocean standard was replaced by the new Ther-modynamic Equation of Seawater 2010 (TEOS-10; IOC et al., 2010). However, as the composition of solutes differs greatly from the ocean, density calculation based on ocean conditions can only be of limited accuracy. Pawlowicz and Feistel (2012) have considered the application of TEOS-10 (IOC et al., 2010) in several cases different from seawater, correcting the salinity values depending on the composition before applying TEOS-10 (IOC et al., 2010). Bührer and Ambühl (1975) developed an equation to calculate density based on temperature and specific conductance at 20◦C for alpine lakes. In addition, a popular approach was formulated by Chen and Millero (1986) for tuning ocean approaches to freshwater conditions (salinity < 0.6 psu).

Higher accuracy can be achieved when site-specific den-sity equations are produced. Jellison et al. (1999) developed a density equation for Mono Lake from water samples which have been measured at different temperatures and dilutions. In the case of meromictic lakes, strong differences in the composition of the mixolimnion and monimolimnion must be reflected in the density equations in order to sustain the permanent stratification in the density calculations. Boehrer et al. (2009) and von Rohden et al. (2010) used an equation

based on density measurements of the monimolimnion and mixolimnion of the Waldsee.

Boehrer et al. (2010) evaluated the contribution of the dif-ferent cations and anions separately in terms of partial mo-lal volumes and implemented an algorithm, RHOMV (http: //www.ufz.de/webax), to calculate density with a second-order approximation for temperature dependence and ionic strength dependence. Pawlowicz et al. (2011) implemented the LIMBETA method that calculates density from composi-tion. Another approach comes from Pawlowicz et al. (2012) where the authors propose to use TEOS-10 but replace seawater salinity with specific salinities obtained and cor-rected for freshwaters. This limnic salinity can be calcu-lated using the chemical composition by summing up all the dissolved solutes (Sasoln) or by summing up only the dissolved ions (Saionic) and correcting this value with the dissolved Si(OH)4, Sadens = Saionic + 50.6 × [Si(OH)4] (mol kg−1). Based on partial molal volumes (RHOMV), Di-etz et al. (2012) separated the contributions of solutes for freshwater lakes. Moreira et al. (2011) based density on the composition of solutes in their model to reproduce the permanent stratification of the Waldsee numerically using RHOMV to include the reactivity of substances in the density (see also Nixdorf and Boehrer, 2015).

These prior studies therefore highlight the necessity of in-cluding the chemical composition to obtain an accurate cal-culation of density. However, we accept the need for a practi-cal density approach, which can easily be implemented, such as a mathematical formula that relates density to temperature and electrical conductivity. In this manuscript, we develop coefficients for such a formula from the chemical composi-tion. We provide an algorithm – RHO_LAMBDA (from ρλ) – to obtain such coefficients and demonstrate the applicabil-ity of the approach with water from Rappbode Reservoir. We also provide an appropriate assessment for the Rapp-bode Reservoir case and compare the accuracy with other approaches currently in use for limnic waters. For a quantita-tive judgement of the general applicability of our approach, we also evaluate coefficients for two further freshwater bod-ies (Lake Geneva, Lake Constance), an extremely saline lake (Mono Lake), a meromictic open pit lake in the mixolimnion and the monimolimnion (Waldsee), and finally seawater as a globally known example and well-defined standard.

2 Methods: the proposed approach (RHO_LAMBDA)

We propose a simple equation for density as a numerical ap-proximation of the (potential) density of lake water:

ρ ≈ ρλ(T , κ25) = ρw(T ) + κ25λ0+λ1× T −25◦C
 , (1)

where the first term of the right side ρwis the density of pure water, which can be calculated in a very accurate way using Kell (1975) or Tanaka (2001). Our approach (Eq. 1) corre-lates density with temperature (T ) and electrical

conductiv-ity at 25◦C (κ25)of a water sample using coefficients λ0and λ1. The introduction of λ1reflects temperature dependence of the density contribution of the solutes, which is required for a shifting temperature of maximum density. Only two co-efficients need to be determined, and thus this equation is easy to implement. Coefficients λ0and λ1can be obtained as follows.

At T = 25◦C, the λ1term in Eq. (1) vanishes and λ0can be calculated using Eq. (2) provided that the water density at 25◦C is known from other sources:

λ0=

ρ (T =25◦C, κ25) − ρw(T =25◦C) κ25

. (2)

If the density is also known for a temperature T 6= 25◦C, λ1 can be calculated in a second step:

λ1=

[ρ (T , κ25) − ρw(T )] /κ25−λ0

T −25◦_C . (3)

Necessary data for Eqs. (2) and (3) can be derived from mea-surements or from calculations.

In the remaining part of this manuscript, Eq. (1), com-plemented by Eq. (2) and Eq. (3), will be referenced as the RHO_LAMBDA approach.

In our RHO_LAMBDA approach, we use the Tanaka (2001) equation for pure water density, ρw. If the compo-sition of solutes in the water is known, the density of water is calculated by using RHOMV (Boehrer et al., 2010) and fi-nally κ25 is provided by the algorithm implemented in the PHREEQC code (Parkhust and Appelo, 1999), whose de-scription can be found in Atkins and de Paula (2009) and on Appelo’s webpage of PHREEQC (Appelo, 2016). This method (re-implemented in Python from the original code) calculates the specific conductance of a solution from the concentration, the activity coefficient and the diffusion co-efficient of all the charged species. The diffusion coco-efficients can be found in Millero (2001).

Rappbode Reservoir

We demonstrate our density approach with the example of Rappbode Reservoir (Germany; for details on this reser-voir, see Rinke et al., 2013, and references therein); its low electrical conductivity indicates low concentrations of so-lutes. From chemical analysis of a surface sample from 19 November 2010, we knew the major cations were calcium (13.8 mg L−1)and sodium (9.3 mg L−1), while major anions were bicarbonate (28.07 mg L−1), sulfate (18.5 mg L−1)and chloride (16.8 mg L−1)(see Table 1). In addition, consider-able portions of organic matter (3.1 mg DOC L−1)and sili-cate (4.5 mg L−1of Si(OH)4)were contained in the sample. The procedure to apply the RHO_LAMBDA method in the case of Rappbode Reservoir can be summarized as follows.

1. For this sample, an electrical conductance κ25= 0.1635 mS cm−1was calculated by inserting given con-centrations into the PHREEQC algorithm (Parkhust and Appelo, 1999; Atkins and de Paula, 2009).

2. According to RHOMV, the density of this sample at 25◦_{C was ρ}

MV(T =25◦C) = 997.130 kg m−3 and ρw(T =25◦C) = 997.047 kg m−3.

3. Putting these numbers into Eq. (2) delivered λ0= 0.506 kg cm m−3mS−1.

4. Similarly, we evaluated λ1= −0.0012 kg cm m−3 mS−1K−1by putting ρw(T =5◦C)= 999.967 kg m−3 and ρMV(T =5◦C) = 1000.053 kg m−3into Eq. (3).

5. Finally, inserting the lambdas as coefficients into Eq. (1) delivered a density formula for Rappbode Reservoir.

3 Assessments

The practicability of this approach depends on its accuracy. This will first be assessed for Rappbode Reservoir water and its above-evaluated coefficients. However, for limnologists working on other limnic water bodies, an assessment of ac-curacy in the general range of limnic water composition is of fundamental interest. In conclusion, we chose a collection of lake waters of different chemical compositions and a wide range of concentrations. We included all lakes we knew of, where a reliable reference density could be provided and the chemical composition was known.

In particular, we included two further typical freshwater lakes, Lake Geneva and Lake Constance, which are well known in the limnological literature. As an example of saline lakes, we chose Mono Lake (e.g. Jellison et al., 1999). In or-der to include also water with a rather unusual composition, we chose two water samples from a meromictic open pit lake, the Waldsee, which contains large amounts of sulfate and dis-solved iron (e.g. Dietz et al., 2008, 2012; Boehrer et al., 2009; von Rohden et al., 2010; Moreira et al., 2011). Finally, we used seawater, of which the composition is known at high accuracy, as a reference water for a standard comparison. Ta-ble 1 presents the original chemical compositions of the dif-ferent lakes considered in the testing of the RHO_LAMBDA expression. Data were derived from chemical analysis (for experimental details, see Appendix A) or literature (for ref-erences, see Table 1). We complemented the set using syn-thetically produced lake water of differing compositions and concentrations from the work by Gomell and Boehrer (2015) in order to test systematically the influence of composition and concentration on the values of the coefficients λ0and λ1 (for experimental details, see Appendix A).

For critical comparison with other density equations, this assessment section (Sect. 3) consists of two major parts: firstly we check the accuracy for different lakes and water samples and secondly we provide the lambda coefficients of several aquatic systems where we have direct measurements or a specifically obtained approach to density (e.g. Mono Lake or seawater) to check the accuracy of ρλ in general.

0 5 10 15 20 25 30 995 996 997 998 999 1000 1001

De

ns

ity

[

kg · m − 3

]

Rappbode Reservoir

0 5 10 15 20 25 30 40 200 20 40 60 80 100

Re

lat

ive

er

ro

r [

%

]

(a)(a)(a)(a)(a)

Reference

RHO_LAMBDA TEOS-10Chen & Millero Buehrer & Ambuehl

0 5 10 15 20 25 30 996.0 996.5 997.0 997.5 998.0 998.5 999.0 999.5 1000.0 1000.5

De

ns

ity

[

kg · m − 3

]

Lake Geneva

0 5 10 15 20 25 30 60 50 40 30 20 100 10 20 30

Re

lat

ive

er

ro

r [

%

]

(b)(b)(b)(b)(b) Reference

RHO_LAMBDA TEOS-10Chen & Millero Buehrer & Ambuehl

0 5 10 15 20 25 30 996.0 996.5 997.0 997.5 998.0 998.5 999.0 999.5 1000.0 1000.5

De

ns

ity

[

kg · m − 3

]

Lake Constance

0 5 10 15 20 25 30 50 40 30 20 100 10 20

Re

lat

ive

er

ro

r [

%

]

(c)(c)(c)(c)(c) Reference

RHO_LAMBDA TEOS-10Chen & Millero Buehrer & Ambuehl

0 5 10 15 20 25 30

Temperature [

◦_C

]

1040 1045 1050 1055 1060 1065 1070 1075 1080

De

ns

ity

[

kg · m − 3

]

Mono Lake

0 5 10 15 20 25 30

Temperature [

◦_C

]

50 40 30 20 10 0

Re

lat

ive

er

ro

r [

%

]

(d)(d)(d)(d) Reference

RHO_LAMBDA TEOS-10 UNESCO

Figure 1.

Table 2 presents the results of the intermediate step calcula-tions to obtain λ0and λ1. As references for the assessment, we used the measured (for details, see Appendix A) or pub-lished data (Fig. 1a and b, Table 2).

The quantitative comparison between the different meth-ods (including the method presented here, RHO_LAMBDA) and the reference values is shown in Fig. 1. Our approach mainly aimed at representing the density contribution of so-lutes. Hence, we related the difference to our reference with the contribution of the solutes

Rel.Error = (ρλ−ρref) / (ρref−ρw) . (4) For temperatures in the range of 1–30◦C, usually found in typical limnic conditions, the values of the relative error

de-fined by Eq. (4) are displayed in the right column of Fig. 1. On purpose, we obtained the chemical composition from a different source (sample) than the density measurement. In this way, the variability of the water composition within one lake was included in the error determination in our assess-ment.

To judge the accuracy of our approach, we also inserted results from other formulas in common use for transferring CTD data into density: we included UNESCO (Fofonoff and Millard, 1983), TEOS-10 (IOC et al., 2010), Chen and Millero (1986) and Bührer and Ambühl (1975) (Fig. 1) as far as possible according to the defined range of applicability of the single formula.

0 5 10 15 20 25 30 995 996 997 998 999 1000 1001

De

ns

ity

[

kg

·

m

− 3

]

Mixolimnion of Lake Waldsee

0 5 10 15 20 25 30 60 50 40 30 20 100 10

Re

lat

ive

er

ro

r [

%

]

(e)(e)(e)(e)(e)

Reference

RHO_LAMBDA TEOS-10Chen & Millero UNESCO

0 5 10 15 20 25 30 996 997 998 999 1000 1001

De

ns

ity

[

kg

·

m

− 3

]

Monilimnion of Lake Waldsee

0 5 10 15 20 25 30 70 60 50 40 30 20 100 10

Re

lat

ive

er

ro

r [

%

]

(f)(f)(f)(f)(f) Reference

RHO_LAMBDA TEOS-10Chen & Millero UNESCO

0 5 10 15 20 25 30

Temperature [

◦_C

]

1021 1022 1023 1024 1025 1026 1027 1028 1029

De

ns

ity

[

kg

·

m

− 3

]

Seawater

0 5 10 15 20 25 30

Temperature [

◦_C

]

3 2 1 0 1 2 3 4

Re

lat

ive

er

ro

r [

%

]

_(g)(g)(g)(g) Reference

RHO_LAMBDA TEOS-10 UNESCO

Figure 1. Test cases. The left panel presents the density curves of the different methods and the right panel shows the relative error of the density contribution of the solutes with respect to the reference. In all the cases, measured values have been used as a reference, except in the cases of Mono Lake (Jellison et al., 1999) and seawater (TEOS-10, IOC et al., 2010), which use specific density equations. (a) Rappbode Reservoir, (b) Lake Geneva, (c) Lake Constance, (d) Mono Lake. Continuation. (e) Mixolimnion of Waldsee, (f) Monimolimnion of Waldsee, (g) seawater.

Rappbode Reservoir. The measured conductance (κ25)of our Rappbode Reservoir sample was 0.1579 mS cm−1, which differed only by 4 % from the value 0.1635 mS cm−1 calcu-lated using the PHREEQC electrical conductivity algorithm at 25◦C (described in Atkins and De Paula, 2009). This is within the measurement accuracy of the chemical analysis. Reference density was produced by measuring in a PAAR DSA 5000 densitometer from 1 to 30◦_C.

We can see that the RHO_LAMBDA method reproduced the reference values of the water sample from Rappbode Reservoir with a relative error ranging from −12.7 to 4.3 %. The deviation from the reference was lower than 5 % in the range 10 to 27◦C. Among the other compared approaches, TEOS-10 showed the best results, with relative error rang-ing from −15.7 to 0.1 %. The Bührer and Ambühl (1975) approach resulted in a relative error ranging from −4.0 to 99.3 % and strongly rising with temperatures increasing above 20◦C. Results according to Chen and Millero (1986) ranged between −37.8 and −25.3 %.

Lake Geneva. Calculated and measured electrical conduc-tivity (κ25)of a water sample from 7 November 2013 differed by less than 1 % for 25◦C (Table 2). Reference density was produced from this sample in a PAAR DSA 5000 densito-meter. The relative error ranged from −11.5 to −4.6 % for our RHO_LAMBDA approach. Bührer and Ambühl (1975) (relative error −15.3 to 21.7 %), TEOS-10 (relative error −12.9 to −7.8 %) and Chen and Millero (1986) (relative er-ror −50.9 to −47.6 %) showed larger deviations from the ref-erence.

Lake Constance. The composition shown in Table 1 mainly coincided with the analysis done by Stabel (1998). The calculated conductivity at 25◦C (κ25)of 0.330 mS cm−1 differed from the measured value of 0.322 mS cm−1by 3 %. The reference density was again from measurements in a PAAR DSA 5000 densitometer. The relative error ranged from −9.7 to −4.2 % for the RHO_LAMBDA approach. TEOS-10 (relative error −12.2 to −8.6 %), Bührer and Am-bühl (1975) (relative error −17.4 to 14.6 %) and Chen and

Table 1. Original chemical composition of the water in different test cases presented in Sect. 3. All values except pH are presented in mg L−1 (NA – not analysed).

Reservoir Lake Lake Mono Waldsee Waldsee Seawater Rappbode Geneva Constance Lake (mixo.) (monimo.)

pH 7.14 7.0 7.9 9.8. 7.1 6.7 7.0 Na+ 9.30 11.1 5.60 32 933.18 9.66 10.81 10 919.56 K+ 1.00 1.74 1.48 1610.92 7.04 10.17 404.23 Ca2+ 13.8 44.30 51.2 6.01 61.32 89.78 417.38 Mg2+ 3.30 6.49 9.03 31.59 12.88 17.50 1299.88 NH+₄ 0.03 < 0.010 < 0.010 NA NA NA NA Fe NA < 0.025 < 0.01 NA 0.50 131.79 NA Fe2+ 0.00 NA NA NA 0.22 117.83 NA Fe3+ NA NA NA NA 0.28 13.96 NA Mn2+ 0.004 < 0.010 < 0.007 NA 0.22 0.88 NA Al3+ 0.01 < 0.005 < 0.02 NA NA NA NA F− 0.00 NA NA NA NA NA 1.31 Cl− 16.8 10.39 7.81 19 043.74 5.67 4.96 19 598.77 SO2−₄ 18.5 44.56 33.26 10 912.64 184.44 176.75 2747.05 NO−₃ 6.50 0.483 3.37 0.00 2.85 1.24 NA HCO−₃ 28.07 94.58 136.68 3276.67 57.97 374.04 106.15 CO2−₃ NA 0.000 NA 17 726.95 0.00 0.00 14.53 Si(OH)4 4.50 0.582 4.42 NA NA NA NA B(OH)−₄ NA NA NA 752.45 NA NA 4.23 DOC density 0.0 0.0 0.0 0.0 0.015 0.060 0.0 correction (kg m−3)* Correction** 0 % −16 % 10.4 % 15 % 7 % −15 % 0 % Data Measured Measured Measured Jellison et al. (1999) Dietz et al. (2008) Millero et al. (2008)

sources Dietz et al. (2012)

* Density modified by addition of this quantity expressed in kg m−3

* Correction of cation concentrations for charge balance

Millero (1986) (relative error −46.6 to −44.1 %) again had larger deviations from the reference. The strong increase in the relative error of Bührer and Anbühl (1975) with temper-ature was smallest for Lake Constance compared to the other freshwater lakes.

Mono Lake. We evaluated density for a water sample of conductivity κ25=85.67 mS cm−1 which was provided by Jellison et al. (1999) and which differed by 12 % from the calculated value 96.61 mS cm−1using the PHREEQC algo-rithm. The density formula by Jellison et al. (1999) was used as the reference density. In this case, the relative error using RHO_LAMBDA ranged from −9.5 to −1.5 % even in this lake with such saline waters and unusual composition. Also in this case, TEOS-10 showed larger deviation from the ref-erence (relative error −10.4 to −5.2 %). The largest relative error was found for the UNESCO equation according to Fof-fonof and Millard (1983) (relative error −39.9 to −36.0 %).

Waldsee mixolimnion/Waldsee monimolimnion. This case presented a meromictic open pit lake (Boehrer et al., 2008; Dietz et al., 2008, 2012; von Rohden et al., 2010; Moreira et al., 2011) of moderate salinity (0.22 psu in the mixolimnion

and 0.6 psu in the monimolimnion, Moreira et al., 2011), but its composition differed from the usual carbonate or chloride waters. Composition was obtained from Dietz et al. (2008, 2012). The DOC (dissolved organic carbon) contribution was added according to Dietz et al. (2012). This correction in-creased density by 0.015 kg m−3in the mixolimnion and by 0.06 kg m−3in the monimolimnion.

The calculated κ25 differed by 7.0 % from the reference value in the mixolimnion and by 7.6 % in the monimolimnion (Table 2). This was the highest difference between refer-ence and calculated values of all waters considered in this study. Probably, the very special chemical composition of the waters was the reason. The missing data for ammonia and silicate may also have contributed, in particular in the monimolimnion. Measurements in the work of Boehrer et al. (2009) were used as a density reference.

The relative error of the RHO_LAMBDA approach ranged from −8.4 to −3.9 % in the mixolimnion. In the moni-molimnion, the relative error ranged from −11.9 to −9.8 % for the RHO_LAMBDA approach. The deviation from the reference was substantially larger for all other compared

ap-Table 2. Summary of the calculated values for obtaining RHOMV_LAMBDA coefficients. Lambdas and references: λ₀and λ₁represent the lambda values obtained using the chemical composition of Table 1 to calculate density and conductivity at 5 and 25◦C, while λ∗₀and λ∗₁ represent the lambda values obtained from a linear regression of the density reference. λ0and λ∗₀are expressed in kg cm m−3mS−1and λ1

and λ∗₁are expressed in kg cm m−3mS−1K−1. Density values are expressed in kg m−3. Corrected salinity shows the values of Practical salinity corrected by a factor of 1.00488 for the Chen and Millero (1986) method. For more details, see text.

Reservoir Lake Lake Mono Waldsee Waldsee Seawater

Rappbode Geneva Constance Lake (mixo.) (monimo.)

κ₂₀ (measured) (µs cm−1) 142.0 263 302 NA NA NA NA κ₂₅ (measured) (µs cm−1) 157.9 294 333.7 85 668 550 1050 53 064.9 κ₂₅(calc.) (µs cm−1) 163.49 296.81 329.77 96 609.50 588.50 969.93 53 762.53 Practical salinity 0.0770 0.127 0.1613 83.04 0.220 0.60 35.00 Absolute salinity 0.0997 0.22 0.251 91.11 0.351 0.78 35.165 Corrected salinity 0.0774 0.128 0.1621 83.45 0.221 0.6029 35.171 ρ_MV (T = 25◦C) 997.130 997.222 997.252 1075.480 997.383 997.744 1023.662 ρ_MV (T = 5◦C) 1000.053 1000.149 1000.181 1083.661 1000.316 1000.691 1028.150 ρref (T = 25◦C) 997.126 997.228 997.253 1069.936 997.391 997.958 1023.344 ρ_ref (T = 5◦C) 1000.059 1000.168 1000.194 1075.447 1000.332 1000.923 1027.600

Data sources Measured Measured Measured Jellison et al. (1999) Dietz et al. (2008) Millero et al. (2008)

Dietz et al. (2012) von Rohden et al. (2010) Moreira et al.(2011) λ₀ 0.51 0.59 0.62 0.81 0.60 0.78 0.50 λ₁ −0.0012 −0.0013 −0.0014 −0.0027 −0.0011 −0.0014 −0.0015 λ∗₀ 0.50 0.62 0.64 0.85 0.63 0.87 0.50 λ∗₁ −0.0042 −0.0034 −0.0033 −0.0015 −0.0020 −0.0022 −0.0013 A.E.1 4.3 % 6.9 % 5.9 % 5.5 % 5.2 % 10.4 % 0.4 % M.E.2 12.7 % 11.5 % 9.7 % 9.5 % 8.4 % 11.85 % 0.75 %

1_{Average absolute value of relative error.}

2_{Max. absolute value of relative error.}

proaches (Fig. 1a and b). The averages of the absolute values of the relative error were 22.8, 52.0 and 52.3 % for TEOS-10 (IOC et al., 2010), Chen and Millero (1986) and UNESCO (Foffonof and Millard, 1983) in the mixolimnion, respec-tively. In the monimolimnion, the values were 35.2 % for TEOS-10, 60.0 % for Chen and Millero (1986) and 60.2 % for UNESCO (Foffonof and Millard, 1983).

Seawater. The seawater composition was obtained from Millero et al. (2008) and we used TEOS-10 (IOC et al., 2010) as our seawater density reference. Electrical conduc-tivity was calculated for this composition and resulted in 53.76 mS cm−1, while the reference value given by Millero et al. (2008) was 53.06 mS cm−1. That meant the deviation was 1.3 %. As expected – both formulas were specifically

de-signed for ocean water –, the relative error of the UNESCO approach according to Foffonof and Millard (1983) was very small, ranging between −0.02 and −0.01 %. This was prob-ably a result of numerical uncertainties of the calculations. The relative error of our RHO_LAMBDA approach ranged between −0.75 and 0.68 %.

4 Discussion

In all cases, our density approach reproduced the density contribution of the salts to within 10 %. This is better than most of the other approaches, which differed by up to 60 % from the correct values. Even in the case of very low concen-trations (Rappbode Reservoir) and very high concenconcen-trations

(Mono Lake) as well as in very special water composition (mine lake Waldsee), the 10 % accuracy for the salt contribu-tion was achieved with our RHO_LAMBDA approach. The observed strong increase in the relative error with temper-ature for Bührer and Ambühl (1975) was caused by its va-lidity limited to 24◦C. Spatial and temporal variability of so-lute composition could contribute to errors in density calcula-tion. However, where we attained chemical composition sep-arately (i.e. from another sample) from the density informa-tion, this error is intrinsically included in our assessment and hence in the value that we supply for the RHO_LAMBDA approach.

The first coefficient λ0 varied by more than a factor of 2 between 0.37 and 0.88 kg cm m−3mS−1; see Fig. 2. This explained that a density formula with constant coefficients could never be able to mimic density accurately for a larger range of lake waters. Obviously the coefficient λ0depended on the composition of the solutes. A dominance of double-charged ions – as opposed to single-double-charged ions – led to higher values of λ0. This effect was clearly visible in the in-clusion of calculated values for a NaCl solution of 1 g L−1 and a CaSO4solution of 1 g L−1(Fig. 2).

Also, the concentration of solutes had a decisive ef-fect on the coefficients. We used density measurements of a dilution series of synthetic lake waters by Gomell and Boehrer (2015) of 1, 3, 10, 30, or 90 g L−1of a mixture of KCl, NaHCO3, and Na2SO4. We included lambda coeffi-cients from RHO_LAMBDA approach “Mix” together with regressions of published measured data “Mix-M” (Fig. 2). Both empirical data as well as RHO_LAMBDA results re-flected the concentration effect on λ0of a factor of 1.5. Al-though not perfect, the agreement between empirical data and RHO_LAMBDA values lay within the 10 % margin we found for lake waters above.

Values for λ1 nearly all lay between −0.001 and −0.002 kg cm m−3mS−1K−1. Hence, the λ1term delivered a small contribution in all cases, i.e. always an order of mag-nitude smaller compared to the λ0term. As a consequence, it could be neglected for most limnological applications. Though not really necessary for an absolute density calcu-lation, λ1was included to also represent the shift of tempera-ture of maximum density for a given lake water composition, which could not be achieved with the λ0term alone. Nega-tive values of λ1indicated a shift of the temperature of max-imum density to lower temperatures. A closer look at the λ1 values revealed that some empirical values (also the Mono Lake reference derived from empirical measurements) were considerably lower than expected from coefficients of physi-cal chemistry. However, the difference posed the question of how accurately the shift of temperature of maximum density would actually be indicated by coefficients of the physical chemistry literature. The largest discrepancies appeared for freshwater lakes (Rappbode Reservoir, Lake Geneva, Lake Constance) where the shift is small.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 λ0(kg·cm/(m3·mS)) 0.006 0.004 0.002 0.000 0.002 0.004 0.006 λ1 (k g · cm /( m 3·m S · K )) Rappbode Constance Mono Seawater Mixol. Waldsee Moni. Waldsee Geneva Rappbode-M Constance-M Mono-M Seawater-M Mixol. Waldsee-M Moni. Waldsee-M Geneva-M NaCl CaSO4 Mix1 Mix3 Mix10 Mix30 Mix90 Mix1-M Mix3-M Mix10-M Mix30-M Mix90-M

Figure 2. Distribution of the values of λ1vs. λ0. Concentrations for

NaCl and CaSO4are 1 g L−1in both cases. Chemical compositions

for the lakes and seawater are presented in Table 1. The water sam-ples labelled as “Mix” are proportional mixtures of KCl, NaHCO3,

and Na2SO4of 1, 3, 10, 30 and 90 g L−1(Gomell and Boehrer,

2015) and the water samples labelled as “-M” correspond to the lambda coefficients obtained from direct measurements of density and conductivity.

The values of λ0and λ1have also been calculated using direct measurements of density (starred values λ∗₀and λ∗₁). In the case of λ0only slight differences can be found between the values calculated from chemical composition and from direct measurements of density. However, those differences increase in the case of λ1, as mentioned above.

5 Conclusions

We showed that the correlation between electrical conduc-tivity and density depends strongly on the composition and concentration of solutes. As a consequence, the limnic range cannot be covered with one formula with constant coef-ficients. However, a simple mathematical addition of two terms to a pure water formula is able to represent the den-sity contribution of solutes in all our examples with an error of less than 10 %. This is sufficient for most limnological ap-plications and is better than any other density approach based on CTD data, if not specifically designed for a given lake wa-ter.

Only two coefficients λ0 and λ1 need to be evaluated: while λ0 varies considerably between lakes, the numeri-cal evaluation of λ1 delivers very similar values of λ1∼ −0.0015 kg cm m−3mS−1K−1for any lake water composi-tion. Hence, once λ0has been evaluated for a lake, a rather

accurate and simple density formula can be used for CTD data. The approach uses conductance κ25, which can be mea-sured in limnic waters, and thus avoids salinity, which is badly defined for limnic waters and, hence, is a precarious quantity. The inclusion of this simple and more accurate ap-proach for potential density calculation in numerical lake models is therefore recommended.

For convenient use and implementation, a density cal-culator tool is provided at https://sourceforge.net/projects/ densitycalc.

6 Data availability

Webax-Web access to numerical tools of limnology is avail-able at http://www.ufz.de/webax. TEOS-10 original Fortran 90 library and related tools are available at http://www. teos-10.org/software.htm. Density calculator is available at https://sourceforge.net/projects/densitycalc. All the data used in this manuscript to calculate the lambda coefficients and density values are provided in Tables 1 and 2.

Appendix A: Measurement of samples taken for this study

All samples were taken as surface samples and stored cooled and without bubbles in polyethylene bottles until measure-ments and analysis in the lab.

Density measurements were done in 1◦C steps between 1 and 30◦C using a PAAR DSA500 densitometer. Measure-ments of electrical conductivity were done with a MultiLab-Pilot conductivity meter (WTW, Germany).

pH was measured using a HQ11d pH meter (Hach-Lange, Germany) in the lab. Sulfate (SO2−₄ )and chloride (Cl−)were analysed by suppressed conductivity using an ICS-3000 ion chromatography system (Dionex, Idstein, Germany) and au-tomatically generated potassium hydroxide eluent. Concen-trations of Ca, Mg, Na, K, Al, Fe, and Mn were determined by optical emission spectroscopy with inductively coupled plasma (ICP-OES, Perkin-Elmer, OPTIMA 3000, Germany) (Baborowski, et al., 2011). Acidity and alkalinity were mea-sured by an automatic titrator (Metrohm, Germany). Bicar-bonate and carBicar-bonate were calculated based on acidity, alka-linity and pH using PHREEQC (Parkhust and Appelo, 1999). Nitrate (NO−₃)(DIN_EN_ISO_13395, 1996; Herzsprung, et al., 2005), ammonium (NH+₄) (Krom, 1980; DIN_EN_ISO_11732, 1997), and silicate (Si(OH)4) (Smith and Milne, 1981) were measured by continuous flow analysis (CFA, Skalar, the Netherlands) (Herzsprung et al., 2006).

Fluoride (F−)and borate (B(OH)−₄)were not included into the analyses because they usually are not relevant for density in typical freshwater lakes.

A1 Corrections of original chemical analyses for charge balance

If the charge balance between cations and anions was higher than 5 % or below −5 %, the concentrations of cations were increased or diminished to reach balance by keeping the ra-tios of the cations to each other constant. The following cor-rections were necessary: reduction by 16 % for Lake Geneva, reduction by 10.4 % for Lake Constance, increase by 15 % for Mono Lake, increase by 7 % for mixolimnion, and reduc-tion by 15 % for monimolimnion of the Waldsee.

A2 Application of the TEOS10 algorithm in the assessment

The initial algorithm of TEOS10 according to IOC et al. (2010) was applied only for seawater serving as a ref-erence. In all other cases, the adaptation for limnic systems proposed by Pawlowicz and Feistel (2012) was used since all other systems are limnic. Because the only difference between both algorithms is the calculation of the so-called absolute salinity and the equation for density is the same, “TEOS-” was used in the legends of all diagrams in Fig. 1a and b.

A3 Preparation of synthetic solutions

For systematic investigation of dependencies of coefficients, λ0 and λ1 we prepared solutions of pure NaCl (1 g L−1) and pure CaSO4(1 g L−1)and proportional mixtures of KCl, NaHCO3and Na2SO4having overall concentrations of 1, 3, 10, 30 and 90 g L−1. The water samples are labelled using the chemical formula of the salts (NaCl; CaSO4)and as “MixN” with N being a number indicating the concentration. More details about these prepared solutions can be found in Gomell and Boehrer (2015).

A4 Software

All the density methods have been implemented in Python 2.7 except the TEOS-10 (IOC et al., 2010). For TEOS-10 the original Fortran 90 library has been downloaded from http: //www.teos-10.org/software.htm and compiled using f2py. The generated Python library has been used directly for the calculations using the Python 2.7 scripts. All the results pre-sented in this manuscript can be obtained using the “den-sity calculator” provided at https://sourceforge.net/projects/ densitycalc.

Acknowledgements. We thank Ulrich Lemmin for taking and sending a water sample from Lake Geneva, and Karsten Rinke for a water sample from Lake Constance.

The article processing charges for this open-access publication were covered by a Research

Centre of the Helmholtz Association.

Edited by: M. Hipsey

Reviewed by: three anonymous referees

References

Appelo, C. A. J.: Specific Conductance: how to calculate, to use, and the pitfalls, available at: http://www.hydrochemistry.eu/ exmpls/sc.html, last access: June 2016.

Atkins, P. and de Paula, A.: Physical Chemistry, 9th ed, Oxford Uni-versity Press, 2009.

Baborowski, M., Büttner, O., and Einax, J. W.: Assessment of Wa-ter Quality in the Elbe River at Low WaWa-ter Conditions Based on Factor Analysis, Clean – Soil, Air, Water, 39, 437–443, doi:10.1002/clen.201000373, 2011.

Boehrer, B.: WEBAX – Web access to numerical tools of limnology, available at: http://www.ufz.de/webax (last access: July 2016), 2010.

Boehrer, B. and Schultze, M.: Stratification of lakes, Rev. Geophys., 46, RG2005, doi:10.1029/2006RG000210, 2008.

Boehrer, B., Dietz S., von Rohden, C., Kiwel, U., Jöhnk, K., Nau-joks, S., Ilmberger, J., and Lessmann, D.: Double-diffusive deep water circulation in an iron-meromictic lake, Geochem. Geophy. Geosy., 10, Q06006, doi:10.1029/2009GC002389, 2009. Boehrer, B., Herzsprung, P., Schultze, M., and Millero, F.

J.: Calculating density of water in geochemical lake strat-ification models, Limnol. Oceanogr-Meth., 8, 567–574, doi:10.4319/lom.2010.8.0567, 2010.

Bührer, H. and Ambühl, H.: Die Einleitung von gereinigtem Ab-wasser in Seen, Schweizerische Zeitschrift für Hydrologie, 37, 347–369, doi:10.1007/BF02503411, 1975.

Burchard, H., Bolding, K., and Villarreal, M. R.: GOTM – a gen-eral ocean turbulence model. Theory, applications and test cases, Tech. Rep. EUR 18745 EN, European Commission, 1999. Chen, C.-T. A. and Millero, F. J.: Precise thermodynamic properties

for natural waters covering only the limnological range, Limnol. Oceanogr., 31, 657–662, 1986.

Cole, T. M. and Buchak, E. M.: CE-QUAL-W2: A Two-Dimensional, Laterally Averaged, Hydrodynamic and Water Quality Model Version 2.0, US Army Corps of Engineers, Wa-terways Experiment Station, 1995.

Dietz, S., Seebach, A., Jöhnk, K. D., Boehrer, B., Knoller, K., and Lessmann, D.: Meromixis in mining lake Waldsee, Germany: hy-drological and geochemical aspects of stratification, Verh. Int. Verein. Limnol., 30, 485–488, 2008.

Dietz, S., Lessmann, D., and Boehrer, B.: Contribution of Solutes to Density Stratification in a Meromictic Lake (Waldsee/Germany), Mine Water Environ., 31, 129–137, doi:10.1007/s10230-012-0179-3, 2012.

DIN_EN_ISO_11732: Water quality – Determination of ammo-nium nitrogen – Method by flow analysis (CFA and FIA) and

spectrometric detection (ISO 11732:2005); German version EN ISO 11732:2005, Beuth Verlag, 1997.

DIN_EN_ISO_13395: Water quality – Determination of nitrite ni-trogen and nitrate nini-trogen and the sum of both by flow analysis (CFA and FIA) and spectrometric detection (ISO 13395:1996); German version EN ISO 13395:1996, Beuth Verlag, 1996. Fofonoff, N. P. and Millard Jr., R. C.: Algorithms for

communica-tion of fundamental properties of seawater, UNESCO technical papers in marine science, 44, 1983.

Gal, G., Hipsey, M. R., Parparov, A., Wagner, U., Makler, V., and Zohary, T.: Implementation of ecological model-ing as an effective management and investigation tool: lake Kinneret as a case study, Ecol. Model., 220, 1697–1718, doi:10.1016/j.ecolmodel.2009.04.010, 2009.

Gomell, A. and Boehrer, B.: Accuracy of lake water density calcu-lated from molar volumes: RHOMW, submitted, 2015.

Herzsprung, P., Duffek, A., Friese, K., de Rechter, M., Schultze, M., and von Tuempling, W.: Modification of a continuous flow method for analysis of trace amounts of nitrate in iron-rich sedi-ment pore-waters of mine pit lakes, Water Res., 39, 1887–1895, doi:10.1016/j.watres.2005.02.017, 2005.

Herzsprung, P., Bozau, E., Büttner, O., Duffek, A., Friese, K., Koschorreck, M., Schultze, M., von Tümpling, W., and Wendt-Potthoff, K.: Routine analysis of sediment pore water of high ionic strength, Acta Hydroch. Hydrob., 34, 593–607, doi:10.1002/aheh.200500656, 2006.

Hodges, B. and Dallimore, C.: Estuary, lake and coastal ocean model: ELCOM, v3.0 user manual, Centre for Water Research, Univ. Western Australia (last access: 12 March 2014), 2007. Imberger, J. and Patterson, J. C.: A dynamic reservoir simulation

model – DYRESM: 5, in: Transport models for inland and coastal waters, edited by: Fischer, H., Academic Press, New York, 310– 361, 1981.

Imboden, D. M. and Wüest, A.: Mixing mechanisms in lakes, in: Physics and chemistry of lakes, edited by: Lerman, A., Imboden, D. M., and Gat, J. R., Springer, Berlin, 83–138, 1995.

IOC, SCOR, and IAPSO: The international thermodynamic equa-tion of seawater – 2010: Calculaequa-tion and use of thermody-namic properties, Manual and Guides No. 56. Intergovernmental Oceanographic Commission, UNESCO (English), available at: http://www.TEOS-10.org (last access: July 2016), 2010. Jellison, R., Macintyre, S., and Millero, F. J.: Density and

conductivity properties of Na-CO3-Cl-SO4 from Mono Lake,

California, USA, International J. Salt Lake Res., 8, 41–53, doi:10.1007/BF02442136, 1999.

Karakas, G., Brookland, I., and Boehrer, B.: Physical character-istics of Acidic Mining Lake 111, Aquat. Sci., 65, 297–307, doi:10.1007/s00027-003-0651-z, 2003.

Katsev, S., Crowe, S. A., Mucci, A., Sundby, B., Nomosatyo, S., Haffner, G. D., and Fowle, D. A.: Mixing and its effects on bio-geochemistry in the persistently stratified, deep, tropical, Lake Matano, Indonesia, Limnol. Oceanogr., 55, 763–776, 2010. Kell, G. S.: Density, thermal expansivity, and compressibility of

liquid water from 0◦ to 150◦C: correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale, J. Chem. Eng. Data, 20, 97–105, doi:10.1021/je60064a005, 1975.

Krom, M.: Spectrophotometric determination of ammonia; a study of modified Berthelot reaction using salicylate and

dichloroiso-cyanurate, Analyst, 109, 305–316, doi:10.1039/AN9800500305, 1980.

McDougall, T. J. and Barker, P. M.: Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox, 28 pp., SCOR/IAPSO WG127, ISBN 978-0-646-55621-5, available at: http://www.teos-10.org/software.htm (last access: July 2016), 2011.

Millero, F. J.: Physical Chemistry of Natural Waters; Wiley-Interscience, New York, 2001.

Millero, F. J., Feistel, R., Wright, D. G., and McDougall, T. J.: The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale, Deep Sea Res. Pt. I, 55, 50–72, doi:10.1016/j.dsr.2007.10.001, 2008.

Moreira, S.: Density Calculator – A water density calculator for lakes, available at: https://sourceforge.net/projects/densitycalc (last access: July 2016), 2014.

Moreira, S., Boehrer, B., Schultze, M., Dietz, S., and Samper, J.: Modelling geochemically caused permanent stratification in Lake Waldsee (Germany), Aquat. Geochem., 17, 265–280, doi:10.1007/s10498-011-9133-4, 2011.

Nixdorf, E. and Boehrer, B.: Quantitative analysis of biogeochemi-cally controlled density stratification in an iron-meromictic lake, Hydrol. Earth Syst. Sci., 19, 4505–4515, doi:10.5194/hess-19-4505-2015, 2015.

Parkhurst, D. L. and Appelo, C. A. J.: Users guide to PHREEQC (version 2). A Computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calcula-tions, Water-Resources Investigation Report 99-4259, U.S. Ge-ological Survey, 1999.

Pawlowicz, R. and Feistel, R.: Limnological applications of the Thermodynamic Equation of Seawater 2010 (TEOS-10), Lim-nol. Oceanogr.-Meth., 10, 853–867, 2012.

Pawlowicz, R., Wright, D. G., and Millero, F. J.: The effects of bio-geochemical processes on oceanic conductivity/salinity/density relationships and the characterization of real seawater, Ocean Sci., 7, 363–387, doi:10.5194/os-7-363-2011, 2011.

Rinke, K., Kuehn, B., Bocaniov, S., Wendt-Potthoff, K., Büttner, O., Tittel, J., Schultze, M., Herzsprung, P., Rönicke, H., Rink, K., Rinke, K., Dietze, M., Matthes, M., Paul, L., and Friese, K.: Reservoirs as sentinels of catchments: the Rappbode Reservoir Observatory (Harz Mountains, Germany), Environ. Earth Sci., 69, 523–536, doi:10.1007/s12665-013-2464-2, 2013.

Rodrigo, M. A., Miracle, M. R., and Vicente, E.: The meromictic Lake La Cruz (Central Spain). Patterns of stratification, Aquat. Sci., 63, 406–416, doi:10.1007/s00027-001-8041-x, 2001. Sanchez España, J., Lopez Pamo, E., Diez Ercilla, M., and

Santofimia Pastor, E.: Physico-chemical gradients and meromic-tic stratification in Cueva de la Mora and other acidic pit lakes of the Iberian Pyrite Belt, Mine Water Environ., 28, 15–29, doi:10.1007/s10230-008-0059-z, 2009.

Smith, J. D. and Milne, P. J.: Spectrophotometric determina-tion of silicate in natural waters by formadetermina-tion of small al-pha, Greek-molybdosilicic acid and reduction with a tin(IV)-ascorbic acid-oxalic acid mixture, Anal. Chim. Acta, 123, 263– 270, doi:10.1016/S0003-2670(01)83179-2, 1981.

Stabel, H. H.: Chemical composition and drinking water quality of the water from Lake Constance, Arch. Hydrobiol. Spec. Issues Advanc. Limnol., 53, 31–38, 1998.

Tanaka, M., Girard, G., Davis, R., Peuto, A., and Bignell, N.: Rec-ommended table for the density of water between 0◦C and 40◦C based on recent experimental reports, Metrologia, 38, 301–309, doi:10.1088/0026-1394/38/4/3, 2001.

Umlauf, L., Burchard, H., and Bolding, K.: General Ocean Tur-bulence Model. Scientific documentation. V3.2, Marine Sci-ence Reports no. 63, Baltic Sea Research Institute Warnemünde, Warnemünde, Germany, 2005.

Vollmer, M. K., Weiss, R. F., Williams, R. T., Falkner, K. K., Qiu, X., Ralph, E. A., and Romanovsky, V. V.: Phys-ical and chemPhys-ical properties of the waters of saline lakes and their importance for deep-water renewal: Lake Issyk-Kul, Kyrgyzstan, Geochim. Cosmochim. Acta, 66, 4235–4246, doi:10.1016/S0016-7037(02)01052-9, 2002.

von Rohden, C., Boehrer, B., and Ilmberger, J.: Evidence for double diffusion in temperate meromictic lakes, Hydrol. Earth Syst. Sci., 14, 667–674, doi:10.5194/hess-14-667-2010, 2010.

Wüest, A., Piepke, G., and Halfmann, J. D.: Combined effects of dissolved solids and temperature on the density stratification of Lake Malawi (East Africa), in: The limnology, climatology and paleoclimatology of the East African Lakes, edited by: Johnson T. C. and Odada, E. O., Gordon and Breach, New York, 183–202, 1996.